Save the bounding box position of an object in a text file
This is an interactive tool, a modal operator that draws the x, y, width and height of the bounding box corners viewed from front (use ortho to check):

import bpy
import bgl
import blf
from mathutils import Vector
from bpy_extras.view3d_utils import location_3d_to_region_2d

class G:
    top_left = Vector()
    bottom_right = Vector()

def draw_poll(ob):
    if ob is not None and ob.type in ('MESH', 'CURVE', 'SURFACE', 'META', 'FONT'):
        return True
    return False

def draw_callback_view(self, context):

    ob = context.object
    if not draw_poll(ob):
        return

    mat = ob.matrix_world

    # 50% alpha, 2 pixel width line
    bgl.glEnable(bgl.GL_BLEND)
    bgl.glColor4f(1.0, 0.0, 0.0, 0.5)
    bgl.glLineWidth(2)

    bbox = [mat * Vector(b) for b in ob.bound_box]

    bgl.glBegin(bgl.GL_LINE_STRIP)
    bgl.glVertex3f(*bbox[0])
    bgl.glVertex3f(*bbox[1])
    bgl.glVertex3f(*bbox[2])
    bgl.glVertex3f(*bbox[3])
    bgl.glVertex3f(*bbox[0])
    bgl.glVertex3f(*bbox[4])
    bgl.glVertex3f(*bbox[5])
    bgl.glVertex3f(*bbox[6])
    bgl.glVertex3f(*bbox[7])
    bgl.glVertex3f(*bbox[4])
    bgl.glEnd()

    bgl.glBegin(bgl.GL_LINES)
    bgl.glVertex3f(*bbox[1])
    bgl.glVertex3f(*bbox[5])
    bgl.glVertex3f(*bbox[2])
    bgl.glVertex3f(*bbox[6])
    bgl.glVertex3f(*bbox[3])
    bgl.glVertex3f(*bbox[7])
    bgl.glEnd()

    min_x = min(b.x for b in bbox)
    max_x = max(b.x for b in bbox)
    min_y = min(b.y for b in bbox)
    max_y = max(b.y for b in bbox)
    min_z = min(b.z for b in bbox)
    max_z = max(b.z for b in bbox)

    bgl.glColor4f(0.0, 1.0, 0.0, 0.5)
    bgl.glBegin(bgl.GL_LINE_STRIP)
    bgl.glVertex3f(min_x, min_y, min_z)
    bgl.glVertex3f(min_x, max_y, min_z)
    bgl.glVertex3f(max_x, max_y, min_z)
    bgl.glVertex3f(max_x, min_y, min_z)
    bgl.glVertex3f(min_x, min_y, min_z)
    bgl.glVertex3f(min_x, min_y, max_z)
    bgl.glVertex3f(min_x, max_y, max_z)
    bgl.glVertex3f(max_x, max_y, max_z)
    bgl.glVertex3f(max_x, min_y, max_z)
    bgl.glVertex3f(min_x, min_y, max_z)
    bgl.glEnd()

    bgl.glBegin(bgl.GL_LINES)
    bgl.glVertex3f(max_x, min_y, min_z)
    bgl.glVertex3f(max_x, min_y, max_z)
    bgl.glVertex3f(min_x, max_y, min_z)
    bgl.glVertex3f(min_x, max_y, max_z)
    bgl.glVertex3f(max_x, max_y, min_z)
    bgl.glVertex3f(max_x, max_y, max_z)
    bgl.glEnd()

    G.top_left = Vector((min_x, min_y, max_z))
    G.bottom_right = Vector((max_x, min_y, min_z))

    bgl.glColor4f(1.0, 0.9, 0.0, 1.0)
    bgl.glPointSize(5)
    bgl.glBegin(bgl.GL_POINTS)
    bgl.glVertex3f(*G.top_left)
    bgl.glVertex3f(*G.bottom_right)
    bgl.glEnd()

    # restore opengl defaults
    bgl.glLineWidth(1)
    bgl.glDisable(bgl.GL_BLEND)
    bgl.glColor4f(0.0, 0.0, 0.0, 1.0)


def draw_callback_px(self, context):

    ob = context.object
    if not draw_poll(ob):
        return

    font_id = 0
    blf.size(font_id, 20, 72)

    region = context.region
    rv3d = context.space_data.region_3d

    text = "%.2f / %.2f" % G.top_left.xz[:]
    dim = blf.dimensions(font_id, text)
    loc = location_3d_to_region_2d(region, rv3d, G.top_left)
    loc.x -= dim[0]
    blf.position(font_id, *loc.to_3d())
    blf.draw(font_id, text)

    text = "%.2f / %.2f" % tuple(map(abs, (G.top_left - G.bottom_right).xz))
    dim = blf.dimensions(font_id, text)
    loc = location_3d_to_region_2d(region, rv3d, G.bottom_right)
    loc.y -= dim[1]
    blf.position(font_id, *loc.to_3d())
    blf.draw(font_id, text)


class ModalDrawOperator(bpy.types.Operator):
    """Draw a line with the mouse"""
    bl_idname = "view3d.modal_operator"
    bl_label = "Simple Modal View3D Operator"

    def modal(self, context, event):
        context.area.tag_redraw()

        if event.type in {'ESC'}:
            bpy.types.SpaceView3D.draw_handler_remove(self._handle_view, 'WINDOW')
            bpy.types.SpaceView3D.draw_handler_remove(self._handle_px, 'WINDOW')
            return {'CANCELLED'}

        return {'PASS_THROUGH'}

    def invoke(self, context, event):
        if context.area.type == 'VIEW_3D':
            # the arguments we pass the the callback
            args = (self, context)
            # Add the region OpenGL drawing callback
            # draw in view space with 'POST_VIEW' and 'PRE_VIEW'
            self._handle_view = bpy.types.SpaceView3D.draw_handler_add(draw_callback_view, args, 'WINDOW', 'POST_VIEW')
            self._handle_px = bpy.types.SpaceView3D.draw_handler_add(draw_callback_px, args, 'WINDOW', 'POST_PIXEL')

            context.window_manager.modal_handler_add(self)
            return {'RUNNING_MODAL'}
        else:
            self.report({'WARNING'}, "View3D not found, cannot run operator")
            return {'CANCELLED'}


def register():
    bpy.utils.register_class(ModalDrawOperator)


def unregister():
    bpy.utils.unregister_class(ModalDrawOperator)

if __name__ == "__main__":
    register()

    for area in bpy.context.screen.areas:
        if area.type == 'VIEW_3D':
            context = bpy.context.copy()
            context['area'] = area
            bpy.ops.view3d.modal_operator(context, 'INVOKE_DEFAULT')
            break


The relevant part for you (front view only):

import bpy
from mathutils import Vector

ob = bpy.context.object
mat = ob.matrix_world
bbox = [mat * Vector(b) for b in ob.bound_box]

min_x = min(b.x for b in bbox)
max_x = max(b.x for b in bbox)
min_z = min(b.z for b in bbox)
max_z = max(b.z for b in bbox)

top_left = Vector((min_x, max_z))
bottom_right = Vector((max_x, min_z))

# x and y
#top_left

# width and height
#tuple(map(abs, (top_left - bottom_right))


And here's some code to write out what you asked for (front view):

import bpy
from mathutils import Vector

def main():

    # specify the object
    ob = bpy.context.object

    # change to your desired output location and file name
    filepath = r"D:\temp\bounds_2d.txt"

    scene = bpy.context.scene
    frame_current = scene.frame_current
    frame_start = scene.frame_start
    frame_end = scene.frame_end

    write_bounds_3d(filepath, ob, frame_start, frame_end)

    scene.frame_set(frame_current)


def write_bounds_3d(filepath, ob, frame_start, frame_end):

    with open(filepath, "w") as file: 
        for frame in range(frame_start, frame_end + 1):
            bpy.context.scene.frame_set(frame)
            file.write("%f %f %f %f\n" % get_bounds(ob))

def get_bounds(ob):
    mat = ob.matrix_world
    bbox = [mat * Vector(b) for b in ob.bound_box]

    min_x = min(b.x for b in bbox)
    max_x = max(b.x for b in bbox)
    min_y = min(b.y for b in bbox)
    max_y = max(b.y for b in bbox)

    top_left = Vector((min_x, max_y))
    bottom_right = Vector((max_x, min_y))

    # make flat tuple, x y width height
    return top_left.to_tuple() + tuple(map(abs, (top_left - bottom_right)))

if __name__ == '__main__':
    main()